Asynchronism Induces Second Order Phase Transitions in Elementary Cellular Automata

Cellular automata are widely used to model natural or artificial systems. Classically they are run with perfect synchrony, i.e., the local rule is applied to each cell at each time step. A possible modification of the updating scheme consists in applying the rule with a fixed probability, called the synchrony rate. For some particular rules, varying the synchrony rate continuously produces a qualitative change in the behaviour of the cellular automaton. We investigate the nature of this change of behaviour using Monte-Carlo simulations. We show that this phenomenon is a second-order phase transition, which we characterise more specifically as belonging to the directed percolation or to the parity conservation universality classes studied in statistical physics

Asynchronous cellular automata

This text has been proposed for the Encyclopedia of Complexity and Systems Science edited by Springer Nature and should appear in 2018.International audienceThis text is intended as an introduction to the topic of asynchronous cellular automata. We start from the simple example of the Game of Life and examine what happens to this model when it is made asynchronous (Sec. 1). We then formulate our definitions and objectives to give a mathematical description of our topic (Sec. 2). Our journey starts with the examination of the shift rule with fully asynchronous updating and from this simple example, we will progressively explore more and more rules and gain insights on the behaviour of the simplest rules (Sec. 3). As we will meet some obstacles in having a full analytical description of the asynchronous behaviour of these rules, we will turn our attention to the descriptions offered by statistical physics, and more specifically to the phase transition phenomena that occur in a wide range of rules (Sec. 4). To finish this journey, we will discuss the various problems linked to the question of asynchrony (Sec. 5) and present some openings for the readers who wish to go further (Sec. 6)

Robustesse des systèmes complexes : étude du rôle de l’aléa dans la dynamique des automates cellulaires

This thesis presents a selection of research works on the theme of cellular automata and discrete complex systems. We are interested in the robustness of these systems, mainly through the use of asynchronous or probabilistic rules. We start from the problem of analysing and classifying the dynamics of asynchronous elementary cellular automata. We use techniques from probability theory such as Markov chains and martingales in order to understand the convergence properties of such systems. In the second part, we present models which are more complex locally and we estimate their robustness properties mainly with numerical simulations. Experiments show that the various collective phenomena we observe have responses to perturbations which are difficult to predict from the analysis of their local rule. The third parts consists of the presentation of three inverse problems, where one seeks to find the rules that produce a given behaviour, here a consensus on the state of the cells or the agents. The overall variety of models that are discussed in the thesis allow us to study a variety of models with complementary techniques, thus following the path opened by Alan Turing in his 1952 article on morphogenesis.Ce mémoire présente une sélection de travaux sur les automates cellulaires et les systèmes complexes discrets. Nous nous intéressons au thème de la robustesse de ces systèmes, à travers l'utilisation de règles asynchrones ou probabilistes. Nous partons du problème de l'analyse et de la classification des automates cellulaires élémentaires asynchrones. Différentes techniques issues de la théorie des probabilités, chaînes de Markov et martingales, sont mises en œuvre pour expliquer les propriétés de convergence de ces systèmes. Dans la deuxième partie, nous présentons des modèles plus complexes, pour lesquels l’estimation de la robustesse se fait principalement à l’aide de simulations numériques. Les expériences montrent des phénomènes collectifs variés et des réponses aux changements difficiles à prédire par l’analyse des règles locales. Dans la troisième partie, l’exploration se poursuit en présentant trois problèmes inverses, problèmes où l’on cherche à trouver des règles qui réalisent un « calcul » sous forme d’un consensus entre cellules ou entre agents. L'ensemble des travaux présentés permet de parcourir différents modèles discrets avec des approches complémentaires. Nous creusons ainsi la question posée par Turing en 1952 où celui-ci évoquait un rôle constructif de l'aléa dans son article sur la morphogenèse

Robustesse des systèmes complexes : étude du rôle de l’aléa dans la dynamique des automates cellulaires

This thesis presents a selection of research works on the theme of cellular automata and discrete complex systems. We are interested in the robustness of these systems, mainly through the use of asynchronous or probabilistic rules. We start from the problem of analysing and classifying the dynamics of asynchronous elementary cellular automata. We use techniques from probability theory such as Markov chains and martingales in order to understand the convergence properties of such systems. In the second part, we present models which are more complex locally and we estimate their robustness properties mainly with numerical simulations. Experiments show that the various collective phenomena we observe have responses to perturbations which are difficult to predict from the analysis of their local rule. The third parts consists of the presentation of three inverse problems, where one seeks to find the rules that produce a given behaviour, here a consensus on the state of the cells or the agents. The overall variety of models that are discussed in the thesis allow us to study a variety of models with complementary techniques, thus following the path opened by Alan Turing in his 1952 article on morphogenesis.Ce mémoire présente une sélection de travaux sur les automates cellulaires et les systèmes complexes discrets. Nous nous intéressons au thème de la robustesse de ces systèmes, à travers l'utilisation de règles asynchrones ou probabilistes. Nous partons du problème de l'analyse et de la classification des automates cellulaires élémentaires asynchrones. Différentes techniques issues de la théorie des probabilités, chaînes de Markov et martingales, sont mises en œuvre pour expliquer les propriétés de convergence de ces systèmes. Dans la deuxième partie, nous présentons des modèles plus complexes, pour lesquels l’estimation de la robustesse se fait principalement à l’aide de simulations numériques. Les expériences montrent des phénomènes collectifs variés et des réponses aux changements difficiles à prédire par l’analyse des règles locales. Dans la troisième partie, l’exploration se poursuit en présentant trois problèmes inverses, problèmes où l’on cherche à trouver des règles qui réalisent un « calcul » sous forme d’un consensus entre cellules ou entre agents. L'ensemble des travaux présentés permet de parcourir différents modèles discrets avec des approches complémentaires. Nous creusons ainsi la question posée par Turing en 1952 où celui-ci évoquait un rôle constructif de l'aléa dans son article sur la morphogenèse

Properties of B2B invoice graphs and detection of structures

International audienceIn economy, a major issue is the potential lack of liquidity for settling the debts generated by payment delays among companies. Since this lack may trigger cascading failures, we analyse the interconnection of debts. Settling debts means lowering the systemic risks. We analyse the data of a large economic network from an Italian invoice operator on a one-year span. We compare different methods to detect structures or communities that could be helpful for debt netting algorithms. The structure of such networks is not currently well known. We give hints on how to sort and identify the type of B2B invoice graphs. In particular, we address the possibility to identify relevant communities in such networks

Two-dimensional traffic rules and the density classification problem

Part 2: Regular PapersInternational audienceThe density classification problem is the computational problem of finding the majority in a given array of votes in a distributed fashion. It is known that no cellular automaton rule with binary alphabet can solve the density classification problem. On the other hand, it was shown that a probabilistic mixture of the traffic rule and the majority rule solves the one-dimensional problem correctly with a probability arbitrarily close to one. We investigate the possibility of a similar approach in two dimensions. We show that in two dimensions, the particle spacing problem, which is solved in one dimension by the traffic rule, has no cellular automaton solution. However, we propose exact and randomized solutions via interacting particle systems. We assess the performance of our models using numeric simulations